It can be shown for any system undergoing simple harmonic motion that the displacement at any time is proportional to the acceleration, and that the constant of proportionality is equal to the square of the oscillation frequency, in radians per second. $a=-\omega^2 x$

For a pendulum consisting of a mass on a string and perturbed from vertical by an angle $\theta$, the restoring force is the component of gravity that is not counteracted by the tension in the string, and is thus equal to $F=ma=-mg\sin\theta \approx -mg\theta \approx -mg \frac{x}{L}$ from which we can determine the oscillation frequency to be $\omega=\sqrt{g/L}$.

We often use fits to determine the relationship between two quantities $x$ and $y$. However, in order to choose the correct type of fit (e.g. linear or quadratic fit), we usually need to know ahead of time what the power-law relationship between $x$ and $y$ is (i.e. is it $y=Ax$ or $y=Ax^2$?). In short, in the equation $y=Ax^b$we are trying to determine $b$.

Many spreadsheet programs have the option to apply a power law fit to data, but if not we can find the power law by applying a linear fit to the logarithm of our data values. If we take the logarithm of both sides of the power law equation, we find $\log y = b \log x + \log A$

Thus, by performing a linear fit to $\log y$ vs. $\log x$ we can determine the power law of the relationship between the two quantities.